**Authors:** Yahya Talebi, Sahar Akbarzadeh

### Abstract

Let $S$ be a commutative semigroup with zero. Let $Z(S)$
be the set of all zero-divisors of $S$. We define the annihilator
graph of $S$, denoted by $ANN_{G}(S)$, as the undirected graph whose
set of vertices is $Z(S)^{\ast}=Z(S)-\{0\}$, and two distinct
vertices $x$ and $y$ are adjacent if and only if $ann_{S}(xy)\neq
ann_{S}(x)\cap ann_{S}(y)$. In this paper, we study some basic
properties of $ANN_{G}(S)$ by means of $\Gamma(S)$. We also show
that if $Z(S)\neq S$, then $ANN_{G}(S)$ is a subgraph of
$\Gamma(S)$. Moreover, we investigate some properties of the
annihilator graph $ANN_{G}(S)$ related to minimal prime ideals of
$S$. We also study some connections between the domination numbers
of annihilator graphs and zero-divisor graphs.

Department of Mathematics

Faculty of Mathematical Sciences

University of Mazandaran, Babolsar

Iran

E-mail: ,

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